Is it permissible for spouses to exchange currencies, such that one spouse transfers currency to the other, who then later transfers back an equivalent amount in a different currency?

For the validity of a currency exchange (sarf), mutual possession must occur at the contracting session. If the exchange is between different currencies, it is not valid due to the absence of mutual possession at the contracting session.

However, if this occurs through a loan, it is permissible for the borrower to repay his debt with another currency with the lender's consent, provided that it is at the exchange rate on the day of repayment and that they do not part ways while anything is still owed between them.

If what is happening is that the husband is being authorized to convert the amount to another currency and then return it, there is no harm in that.